Question

find the total number of rectangles in a 8x9 chequered board.

Answer 1

my guess is 72

Answer 2

or do you mean all permutations?

Answer 3

It's not 72.

Answer 4

its not 72. the number is much bigger. this question is under the topic counting/progression. In order to do this you need to find the sequence; i.e. what is no. of rectangles. 1x2, 2x3, 3,4 board.

Answer 5

i figured it wasnt that easy after i posted :) yeah, this is one of those...count the bigger versions of it... type issue

Answer 6

help please! ><

Answer 7

It's a pretty common problem, and am not in the mood to work through it myself, but this gives a pretty good explanation: http://mathforum.org/library/drmath/view/56760.html

Answer 8

its aolt of counting :)

Answer 9

didnt really help.. :/

Answer 10

9C2 x 10C2 isn't a lot of counting.

Answer 11

It's a lot of counting if you apply a medieval method, sure.

Answer 12

anyway. i have another question

Answer 13

find the total number of rectangles in 8x8 board. the answer in the link you gave me is 1296. but i got 196 instead. am i wrong?

Answer 14

I think we can assume you are wrong, yes; you can post an outline of your method for assessment, if you want.

Answer 15

i can explain how i derive mine. by doing the board from 1x1, 2x2,3x3, 4x4, it produces the answer

Answer 16

1x1 -> 0

Answer 17

2x2-> 4
3x3-> 16
4x4-> 36

Answer 18

Firstly, a square is a type of rectangle.

Answer 19

which follows a pattern of \[(2n-2)^{2}\]

Answer 20

my question stats that: think of the best way of specifying a rectangle. therefore i can decide whether to include squares as rectangles.

Answer 21

states*

Answer 22

for my benefit in this case i chose not to.

Answer 23

So it's actually:
1x1 -> 1
2x2 -> 9 (yours, plus the whole thing for 1, plus the 4 1x1s)
3x3 -> 37

The pattern is actually

\[1^3 + 2^3 + 3^3 + ... + ... n^3 \]

Answer 24

Firstly, you cannot decide a square isn't a rectangle. And second, your answer is still wrong.

Answer 25

why is a square a rectangle?

Answer 26

all squares are rectangles; but not all rectangles are squares.

Answer 27

rectangle = opposites sides are the same; and 4, 90 degree corners

Answer 28

There are 1296 rectangles.
There are 204 squares.

There are 1092 rectangles that are not squares.

---

A rectangle is just a 4 sided shape with 4 right angles.

Answer 29

Just a note - there was a typo above -> 3x3 gives 36 not 37.

Answer 30

actually a rectangle is a shape that has 2 equal sides.

Answer 31

which makes square also a rectangle. ok i had that clarified thanks.

Answer 32

Actually, you don't need to say it has two equal sides (and that is not explicit enough - see: parallelogram). I direct you to my definition.

Answer 33

a parallelogram is also a rectangle

Answer 34

No, a rectangle is a parallelogram. The converse is not true - i.e. a parallelogram is not always a rectangle.

Why are you arguing with me? You obviously have no idea who I am. That, and I'm trying to help you.

Answer 35

ok i'm sorry. i googled that and it told me that a parallelogram is a rectangular. guess it came out wrong.

Answer 36

3*3=36. HOW?

Answer 37

*facepalm*

Answer 38

its 1^3 + 2^3 + 3^3

Answer 39

LOL. and newton who are you anyway? ^^

Answer 40

IF IT IS 3^3 THEN ANSWER IS 27 NOT 36???

Answer 41

3x3, in this context, means the number of rectangles on a 3x3 board.

And I'm the person who has never made a Mathematical mistake in their entire life.

Answer 42

at leat no mistakes that anyone has lived to tell about ;)

Answer 43

LOL. are you a lecturer? >.>

Answer 44

YOU R SAYING IT IS NOT A MISTAKE? FUNNY

Answer 45

how do you remove see out of this conversation?

Answer 46

btw newton. can you help me in 8x9 rectangles?

Answer 47

ANY THING WRONG HOLYX?????????????

Answer 48

No, I'm not a lecturer (or anything similar).

And the number of rectangles on an 8 by 8 board can be obtained thus:

There are 10 lines in one direction, and 9 lines in the others. To make a rectangle you need to pick 2 of each, so this is:

\[^10C_2 \times ^9C_2 \text{ where } ^nC_r = \frac{n!}{r!(n-r)!} \]

You can do it by slower method of adding up systematically, but I will not partake in them.

Answer 49

8 by 9*

Answer 50

Ugh, LaTeX fail:

\[^{10}C_2 \]**

Answer 51

you mean 8!/(8-r)!r!

Answer 52

wait im thinking. lol

Answer 53

i dont get it. why 10c2. x 9c2. why do you need to multiply.

Answer 54

Because there are x ways to choose the horizontal lines, and y ways to chose the vertical ones, and therefore x * y ways to combine these.

Say you choose the first set of line horizontally. You can pick any of the sets vertically to go with this. so this would be y ways.

You then move onto the next way to pick vertical lines, and can do each of these with the horizontal ones, so another y ways.

.

You finally get to the xth set of vertical lines, and another y.

Total = x * y

Answer 55

Ugh, I mixed up vertical and horizontal in each part, but hopefully it made sense.

Answer 56

ok yea. 9c2 x 9c2 gives the same aswer 1296!

Answer 57

:)

Answer 58

omg. you're damn smart. how did you thought of that. -,- wtf.

Answer 59

anyway thanks for answering my question (: